Research for implementation of PANS model based on open source code

2.1 Governing Equations of RANS

In RANS method, instantaneous velocity field is decomposed into time averaged component, U_i and fluctuation component u_i like Equation (1).

It is well known that the governing Equation (2) and (3) for the RANS in incompressible flow are as follows;

Continuity equation:

Momentum equation:

To model the unknown Reynolds stress τ_ij, various turbulence models were suggested. In this work, we selected the k-ω turbulence model since it provides better performance than those of RANS simulation [8][9]. This means that unsteady simulation is more appropriate than steady simulation in the fluctuating flow associated with massive separation.

Model equations are shown in Equation (4) and (5).

The model constants and the eddy viscosity are given by

$$ \mathrm{\alpha }=\frac{5}{9},\mathrm{}\mathrm{\beta }=\frac{3}{40},\mathrm{}{\mathrm{\beta }}^{*}=\frac{9}{100}\mathrm{},\mathrm{}{\mathrm{\sigma }}^{*}=0.5\mathrm{},\mathrm{}\mathrm{\sigma }=0.5$$, and $$ {\nu }_t=\frac{k}{\omega }$$

Typical boundary conditions at the wall are specified as

$$ \mathrm{k}=0,\mathrm{}\mathrm{\omega }=\frac{60\upsilon }{\beta {\left(y_1\right)}^2}$$

where y₁is the physical distance of the first grid point away from the wall.

2.2 Governing Equations of PANS

In PANS method, the instantaneous velocity field is decomposed into resolved velocity field, U_i and unresolved velocity field, u_i like Equation (6). The expression of equation is same to Equation (1) but the physical and mathematical meaning is different.

The PANS in incompressible flow are as follows;

Continuity equation:

Momentum equation:

To make the closed system on these equations, additional equations are also necessary and presented in Equation (9) and (10).

Where, $$ \stackrel{-}{\beta }=\left(\alpha {\beta }^{\mathrm{*}}+f_k\left(\beta -\alpha {\beta }^{\mathrm{*}}\right)\right)$$, the model constants and the eddy viscosity are given by

$$ \alpha =\frac{5}{9}$$, $$ \beta =\frac{3}{40}$$, $$ {\beta }^{*}=\frac{9}{100}$$, $$ {\sigma }_{ku}=0.5$$, $$ {\sigma }_u=2.0$$, and $$ {\nu }_u=\frac{k_u}{{\omega }_u}$$.

As shown in Equation (9) and (10), basic formulations are almost similar except for the model coefficient, β^* comparing to Equation (4) and (5). As shown, Equation (11), this coefficient is defined by control parameter, f_k which means the ratio of unresolved to total turbulent quantity.

Where k is turbulent kinetic energy and subscript u means unresolved quantities which can be expressed in modeled equations of k_u. Since PANS [2] has been suggested, many researchers have been investigated how to decide the control parameter[10][11]. In this study, the control parameter, f_k is defined as a constant, 0.5 for a fundamental comparison of two methodologies.

3.1 Target Flow 1

As written previously, the flow past a wall mounted square cylinder is defined as the target flow 1 and this flow have been studied by experiments [4][5]. It is well known that this flow show massive flow separation at corner of a square cylinder. The computational domain is shown in Figure 1, and designed based on the experimental study. The Reynolds number (Re_D) based on the width of the square cylinder (D) and the free stream velocity (U_Ref) is 11,540. The boundary condition is given in Figure 2. Periodic boundary condition was designated on two sides in spanwise direction. The Neumann condition was adopted for the outflow boundary. At all the surface boundaries including a wall mounted square cylinder and bottom surface, no slip condition was imposed, respectively. In inflow boundary, the velocity was defined as a Dirichlet condition. At top, the slip condition was given with same velocity with inflow boundary. The grid system was presented in Figure 3, carried out in previous study[6]. Total number of grid is about 400,000. The minimum ∆z⁺ at the center of the top surface of the building is 1.71.

Figure 1:

Computational domain for the target flow 1

Figure 2:

Boundary condition for the target flow 1

Figure 3:

Grid system for the target flow 1

3.2 Target flow 2

The second target flow was defined as the flow around ship. Especially, the KCS is a famous benchmark hull-form for a numerical study, experimental data were also released [12]. The main particulars and geometry are given in Figure 4 and Table 1, respectively. The simulation was carried out in model scale.

Figure 4:

KCS hull geometry

Table 1:

Specifications for KCS

The computational domain and boundary conditions are shown together in Figure 5. Since the hull is a symmetrical geometry, the half-body is only considered for saving the computational time and cost. As presented in Figure 6, the unstructured grid system was constructed and the total number of cells is about 1,200,000. The validated grid system in previous study [7] was adopted.

Figure 5:

Boundary condition for the target flow 2

Figure 6:

Grid system for the target flow 2

3.3 Code for Numerical Simulation

For numerical simulation, we employed an open source code, called by Open Field Operation and Manipulation (Open-FOAM) which is a one of the well-known incompressible Navier-Stokes solvers. Since this code has been enough verified in many engineering problems, it is thought that there is no doubt about the reliability of the code. The PIMPLE algorithm was implemented for coupling of pressure and velocity, and Crank-Nicholson scheme was used for the temporal discretization. The Gauss linear scheme with second order accuracy was applied for spatial discretization.

4.1 Target Flow 1

As already mentioned, a periodic vortex shedding on this flow is the representative characteristics. Therefore, the transient simulation is unavoidable to get the time averaged flow field which was obtained over 150 non-dimensional time. This time period means about 15~16 vortex shedding cycles in this target flow.

The streamwise velocity (U) profiles at 4 streamwise positions (x/d=1, 2, 4, and 6; y/d=0) are compared in Figure 7. In this wake region, the velocity profiles from RANS model and PANS are similar each other, but those from PANS shows better agreement with experimental data for the region on z/d<3. The vertical velocity (W) profiles at the center plane (y/d=0) are given in Figure 8. We again see that the velocity profiles from RANS and PANS are similar. But PANS shows better performance in the region on z/d<3. This target flow is naturally unstable due to a large scale flow separation. The flow unsteadiness is dominantly induced by the Karman vortex shedding as explained by previous studies. It is also known that the oscillating flow pattern at the half height plane of a long wall mounted square cylinder is very similar to that of the flow around a two-dimensional square cylinder [4]-[6].

Figure 7:

U velocity profile comparison at four locations

Figure 8:

W velocity profile comparison at four locations

To examine the predictability of the flow unsteadiness in two different methods, Figure 9 shows the vorticity with respect to Y direction distribution extracted at different 4 positions (z/d=1, z/d=2, z/d=3,and z/d=4). The unsteadiness of flow is clearly shown in both simulations and the massive flow separation is observed in 3 positions (z/d=1, z/d=2, and z/d=3). From Figure 9, it is known that vortices with small scale are described in PANS model.

Figure 9:

Vorticity distribution in 4 locations of different height

To compare the turbulence flow structure, Q-criteria is shown in Figure 10. In both simulations, the horseshoe vortices in front of the wall mounted square cylinder was observed similarly. But the flow structures behind the obstacle is more realistically examined in PANS simulation. This shows that the PANS has advantages to simulate the turbulence flow field with massive separation.

Figure 10:

Q Criteria of numerical simulations (Q=0.5)

4.2 Target Flow 2

In contrast to previous section, the ship is a representative example of the streamlined body. In streamlined body, the massive separation of flow is generally not observed. Of course two pair of bilge vortex is shown in the wake region behind a ship, but it is directly affected by ship type such as a tanker, container, and LNG carrier. Since the ship speed of a container is relatively larger than other types of vessel, the hull form is designed to be more streamlined to reduce the resistance, these bilge vortex is also very weaken.

The total resistance coefficient, C_TM is compared in Table 2. The results from both simulations showed a similar value and within 1.0% difference. But RANS is slightly agreed well to that of model test rather than PANS.

Table 2:

Resistance of KCS in calm sea condition

The Figure 11 shows the pressure distribution on a FWD-hull by two methods. As well-known, the resistance is oriented by the summation of pressure and skin friction on the target object in general. Since the total resistance value is similar each other, as shown in Table 2, it can be inferred that the pressure distribution on a hull is also similar, regardless of the numerical method. In a container ship, it is well known that the portion of the wave resistance in total resistance is relatively larger than the other ship types for example, tanker, bulker and LNG carrier. Therefore the Kelvin wave around container is clearly observed in general. In both simulations, the predicted Kelvin wave are almost similar, as shown in Figure 12.

Figure 11:

Pressure distribution on KCS

Figure 12:

Free surface distribution on KCS

From results based on the second target flow, the difference between of RANS and PANS is relatively negligible rather than those of first target flow.

Author Contributions

Conceptualization, G.S. Song; Methodology, G.S. Song; Software, G.S. Song, S.W. Lee; Validation, G.S. Song, S.W. Lee; Resources, S.W. Lee; Data Curation, G.S. Song; Writing—Original Draft Preparation, G.S. Song; Writing—Review & Editing, G.S. Song; Visualization, G.S. Song; Supervision, G.S. Song; Project Administration, G.S. Song; Funding Acquisition, G.S. Song.